# The blank in the statement “The solution of the inequality | x | ≤ 3 is the interval _____”.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.7, Problem 3E

(a)

To determine

## To fill: The blank in the statement “The solution of the inequality |x|≤3 is the interval _____”.

Expert Solution

The complete statement is “The solution of the inequality |x|3 is the interval [3,3]_”.

### Explanation of Solution

Property used:

Properties of absolute value inequalities:

 Inequality Equivalent form |x|c x<−c or c

Calculation:

The given inequality is |x|3.

This is of the form |x|c.

Then by the properties mentioned above, the given inequality is equivalent to 3x3.

Note that, here both the end points are included since the inequality sign is .

Then, the solution will be a closed interval. That is, [3,3].

Thus, the complete statement is “The solution of the inequality |x|3 is the interval [3,3]_”.

(b)

To determine

### To fill: The blank in the statement “The solution of the inequality |x|≥3 is a union of two intervals _____ and ____”.

Expert Solution

The complete statement is “The solution of the inequality |x|3 is a union of two intervals (,3][3,)_”.

### Explanation of Solution

The given inequality is |x|3.

This is of the form |x|c.

Then by the properties mentioned in part (a), the given inequality is equivalent to x3or 3x.

That is, the solution of the given inequality is the union of the set of all points that are less than or equal to 3, and the set of all points that are greater than or equal to 3.

Note that, here both the end points are included since the inequality sign is .

The set of all points that are less than or equal to 3, are denoted by the interval (,3] and the set of all points that are greater than or equal to 3 is denoted by the interval [3,).

Thus, the complete statement is “The solution of the inequality |x|3 is a union of two interval, (,3][3,)_”.

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